Fokker-Planck description of the transfer matrix limiting distribution in the scattering approach to quantum transport

The scattering approach to quantum transport through a disordered quasi-one-dimensional conductor in the insulating regime is discussed in terms of its transfer matrix \bbox{T}. A model of $N$ one-dimensional wires which are coupled by random hopping matrix elements is compared with the transfer matrix model of Mello and Tomsovic. We derive and discuss the complete Fokker-Planck equation which describes the evolution of the probability distribution of \bbox{TT}^{\dagger} with system length in the insulating regime. It is demonstrated that the eigenvalues of \ln\bbox{TT}^{\dagger} have a multivariate Gaussian limiting probability distribution. The parameters of the distribution are expressed in terms of averages over the stationary distribution of the eigenvectors of \bbox{TT}^{\dagger}. We compare the general form of the limiting distribution with results of random matrix theory and the Dorokhov-Mello-Pereyra-Kumar equation.Comment: 25 pages, revtex, no figure